Numerical calculations of the $B_{1g}$ Raman spectrum of the two-dimensional Heisenberg model

The $B_{1g}$ Raman spectrum of the two-dimensional $S=1/2\mathrm{}$ Heisenberg model is discussed within Loudon-Fleury theory at both zero and finite temperature. The exact $T=0\mathrm{}$ spectrum for lattices with up to $6\times 6$ sites is computed using Lanczös exact diagonalization. A quantum Monte Carlo (QMC) method is used to calculate the corresponding imaginary-time correlation function and its first two derivatives for lattices with up to $16\times 16$ spins. The imaginary-time data are continued to real frequency using the maximum-entropy method, as well as a fit based on spin-wave theory. The numerical results are compared with spin-wave calculations for finite lattices. There is a surprisingly large change in the exact spectrum going from $4\times 4$ to $6\times 6$ sites. In the former case there is a single dominant two-magnon peak at $\omega /J\approx 3.0$, whereas in the latter case there are two approximately equal-sized peaks at $\omega /J\approx 2.7$ and $3.9$. This is in good qualitative agreement with the spin-wave calculations including two-magnon processes on the same lattices. The spin-wave results for larger lattices show how additional peaks emerge with increasing lattice size, and eventually develop into the well known two-magnon profile peaked at $\omega /J\approx 3.2$ and with weight extending up to $\omega /J\approx 4.6$. Both the Lanczös and the QMC results indicate that the actual two-magnon profile is broader than the narrow peak obtained in spin-wave theory, but the positions of the maxima agree to within a few percent. The higher-order contributions present in the numerical results are merged with the two-magnon profile and extend up to frequencies $\omega /J\approx 7$. The first three frequency cumulants of the spectrum are in excellent agreement with results previously obtained from a series expansion around the Ising limit. Typical experimental $B_{1g}$ spectra for ${\mathrm{La}}_2{\mathrm{CuO}}_4$ are only slightly broader than what we obtain here. The exchange constant extracted from the peak position is $J\approx 1400\hspace{0.5em}\mathrm{K}$, in good agreement with values obtained from neutron scattering and NMR experiments. We discuss the implications of our present results for more sophisticated theories of Raman scattering suggested recently.